I would suspect the trusted locations thing would solve the problem. So this other computer has a full version of Access? Maybe looking into the advanced options of that computer's trusted locations within Access will glean some light. IIRC, there are options for importing and or specifying documents like Word and Excel docs.
How are you importing? Did you use the Wizard to create the button? I would use the Wizard to import a Text file using Advanced options and save that procedure with a name. Then, I would crate a button that uses VBA to call the saved Import procedure.
I believe the TransferText method of the DoCmd Object is still valid ...
Code:
DoCmd.TransferText acImportDelim, "ImportName", "TableName", "FilePathAndName", 0 'Zero indicates that there are not any field names in the file